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Theorem 1st2nd 6385
Description: Reconstruction of a member of a relation in terms of its ordered pair components. (Contributed by NM, 29-Aug-2006.)
Assertion
Ref Expression
1st2nd  |-  ( ( Rel  B  /\  A  e.  B )  ->  A  =  <. ( 1st `  A
) ,  ( 2nd `  A ) >. )

Proof of Theorem 1st2nd
StepHypRef Expression
1 df-rel 4877 . . 3  |-  ( Rel 
B  <->  B  C_  ( _V 
X.  _V ) )
2 ssel2 3335 . . 3  |-  ( ( B  C_  ( _V  X.  _V )  /\  A  e.  B )  ->  A  e.  ( _V  X.  _V ) )
31, 2sylanb 459 . 2  |-  ( ( Rel  B  /\  A  e.  B )  ->  A  e.  ( _V  X.  _V ) )
4 1st2nd2 6378 . 2  |-  ( A  e.  ( _V  X.  _V )  ->  A  = 
<. ( 1st `  A
) ,  ( 2nd `  A ) >. )
53, 4syl 16 1  |-  ( ( Rel  B  /\  A  e.  B )  ->  A  =  <. ( 1st `  A
) ,  ( 2nd `  A ) >. )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   _Vcvv 2948    C_ wss 3312   <.cop 3809    X. cxp 4868   Rel wrel 4875   ` cfv 5446   1stc1st 6339   2ndc2nd 6340
This theorem is referenced by:  2ndrn  6387  1st2ndbr  6388  elopabi  6404  cnvf1olem  6436  ordpinq  8812  addassnq  8827  mulassnq  8828  distrnq  8830  mulidnq  8832  recmulnq  8833  ltexnq  8844  fsumcnv  12549  cofulid  14079  cofurid  14080  idffth  14122  cofull  14123  cofth  14124  ressffth  14127  isnat2  14137  nat1st2nd  14140  homadmcd  14189  catciso  14254  prf1st  14293  prf2nd  14294  1st2ndprf  14295  curfuncf  14327  uncfcurf  14328  curf2ndf  14336  yonffthlem  14371  yoniso  14374  dprd2dlem2  15590  dprd2dlem1  15591  dprd2da  15592  2ndcctbss  17510  utop2nei  18272  utop3cls  18273  caubl  19252  rngoi  21960  drngoi  21987  nvop2  22079  nvvop  22080  nvop  22158  phop  22311  cvmliftlem1  24964  fprodcnv  25299  heiborlem3  26503  isdrngo1  26553  iscrngo2  26589
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-iota 5410  df-fun 5448  df-fv 5454  df-1st 6341  df-2nd 6342
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